A variant of the Berry phase, known as the Zak phase [1]},
obtained by integrating across the Brillouin zone of a crystalline
system, plays the central role in the modern theory of
polarization [2]}, [3]}, [4]}, [5]} (MTP).
The Zak phase is proportional to the bulk polarization of a
crystalline system. It is of interest to note that our understanding
of conduction and insulation in quantum systems was initiated by the
seminal work of Kohn [6]}, who emphasized that the classical
idea of localization of single charge carriers as a criterion of
insulation loses its validity. Instead, the criterion of insulation,
according to Kohn, is many-body localization, the localization of the
center of mass of the entire charge distribution (or at least the
charge distribution of large chunks of the sample). In systems with
open boundaries, Kudinov [7]} showed that the variance of
the polarization (which is proportional to the variance of the center
of mass of the charge distribution) is an appropriate criterion to
distinguish conductors from insulators. For crystalline systems,
however, it was not possible to calculate the center of mass of the
charge distribution, nor its variance or higher cumulants, because the
relevant quantum mechanical operator, the position operator, is
ill-defined under periodic boundary conditions. It was precisely this
problem that was
solved [2]}, [3]}, [4]}, [5]} by MTP, by
casting the polarization in crystallie systems as a Zak phase.
